= Finite-time blowup
{title2=$\lim_{t\uparrow T}\|y(t)\|=\infty,\quad T<\infty$}
A <solution of a differential equation> has <finite-time blowup> in norm when $\|y(t)\|\to\infty$ as $t$ approaches a finite endpoint $T$ of its maximal <interval> of existence. This is stronger than merely approaching an excluded <boundary> of an open state domain while staying bounded. The <finite-dimensional continuation criterion> distinguishes these obstructions; for a locally <Lipschitz continuous> <ordinary differential equation> on $I\times\mathbb R^d$, a finite maximal endpoint inside $I$ necessarily entails <finite-time blowup>.
A simple example with a smooth right-hand side is
$$
y'=1+y^2,\qquad y(0)=0,\qquad y(t)=\tan t,\qquad -\frac\pi2<t<\frac\pi2.
$$
The <derivative> identity $(\tan t)'=1+\tan^2t$ verifies the equation and the <initial condition>. Also $|u^2-v^2|=|u-v||u+v|\leq2R|u-v|$ for $|u|,|v|\leq R$, giving local <Lipschitz continuity> and therefore uniqueness. Yet the positive blowup time is $T=\pi/2\approx1.57079633$. Computed values show the accelerating growth:
$$
y(1)\approx1.55740772,\qquad y(1.4)\approx5.79788372,\qquad y(1.5)\approx14.10141995.
$$
The right panel under the <Picard-Lindelöf theorem> deliberately clips its vertical axis at $10$: the trajectory has already left the picture at $t=\arctan10\approx1.47112767$, before reaching $T$. A plot ceiling is a display limit, not a finite value at blowup.
The <tangent function> also quantifies the last part of the trajectory. With $\delta=T-t>0$,
$$
y(T-\delta)=\frac{\cos\delta}{\sin\delta},\qquad \delta\,y(T-\delta)\longrightarrow1\quad(\delta\downarrow0).
$$
Thus the height is approximately the reciprocal of the remaining time. At remaining times $\delta=0.1,0.01,0.001$, the heights are approximately $9.96664442,99.99666664,999.99966667$. These follow from the <sine> and <cosine> limits at zero and explain the steep rise without relying on the appearance of a clipped <function graph>.
The distinction is useful when interpreting a <solution of a differential equation>: uniqueness certifies a trajectory while it exists, while growth estimates decide how far it can be followed. For $y'=1+y^2$, the <Lipschitz constant> $2R$ on a bounded state <interval> grows with $R$, and the right-hand side grows quadratically. There is no global <Lipschitz constant> or linear-growth bound of the kind used by the <finite-dimensional continuation criterion>. Smoothness at every finite state is entirely compatible with <finite-time blowup>.
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